Sampling method, reconstruction method, and device for sampling and/or reconstructing signals

ABSTRACT

Reconstruction method for reconstructing a first signal (x(t)) regularly sampled at a sub-Nyquist rate, comprising the step of retrieving from the regularly spaced sampled values (y s [n], y(nT)) a set of weights (c n , c nr , c k ) and shifts (t n , t k ) with which said first signal (x(t)) can be reconstructed. 
     The reconstructed signal (x(t)) can be represented as a sequence of known functions (γ(t)) weighted by the weigths (c k ) and shifted by the shifts (t k ). The sampling rate is at least equal to the rate of innovation (ρ) of the first signal (x(t)).

The present invention relates to a sampling method, a reconstructionmethod and related devices for sampling and/or reconstructing signals.

With the increasing use of digital devices in all scientific andtechnical fields, the need for reliable acquisition devices equallyincreases. The role of these devices is to acquire the analog signals orwaveforms of the continuous-time world and convert them in time-discretedigital signals which can then be processed by said digital devices, forinstance to perform some computational operations such as processing,decoding or reconstructing the analog waveform. Such acquisition devicesare used in many areas of science and technology, including for examplescientific measurements, medical and biological signal processing,telecommunication technology, etc.

The acquisition devices sample the analog waveform at uniform samplingintervals and with a regular sampling frequency, thus generating a setof sampled values of the sampled signal. It is critical in most samplingschemes to use the smallest set of representative samples needed tofully represent and eventually allow the faithful reconstruction of theanalog waveform. In other words, the smallest sampling frequency stillallowing a faithful representation of the first signal is the goal.

The sampling and reconstruction methods currently used in commonacquisition devices are based on the sampling theorem of Whittaker,Kotelnikov and Shannon.

This theorem states that a signal, for example a received signal y(t),bandlimited to the frequency band [−ω_(m), ω_(m)], can be completelyrepresented by samples y_(s)[n] spaced by an uniform sampling interval Tif the sampling rate 2π/T is at least twice the bandwidth ω_(m).

This sampling scheme is represented on FIG. 1. The lowest samplingfrequency 2ω_(m) given by this theorem is commonly referred to as theNyquist rate or Shannon frequency. In other words, the minimal samplingfrequency directly depends on the bandwidth of the analog signal y(t) tosample and/or reconstruct.

The reconstruction method related to this sampling theorem allows aperfect reconstruction of a signal y(t) by superposing regularly spacedsin c functions regularly delayed by the value of one sampling intervalT and weighted by the successive sampled values y_(s)[n] of the sampledsignal. This well-known reconstruction scheme is illustrated on FIG. 2,where the bloc 1 illustrates the sin c reconstruction filter.

If the signal y(t) to sample and/or reconstruct is a non-bandlimitedsignal, or if its bandwidth is too high to sample it at an acceptablesampling frequency, one usually filters it with a lowpass filter, thusgenerating a bandlimited lowpass version of the signal. This lowpassversion of the signal can then be sampled and reconstructed with theabove-described sampling and reconstruction schemes. The lowpassfiltering can be performed by a dedicated lowpass filter circuit, by thetransfer function of a transfer channel over which the signal y(t) hasbeen transmitted, and/or by the transfer function of a measuring deviceor demodulating circuit used for acquiring this signal.

Therefore, when sampling and reconstructing a non-bandlimited signaly(t) or a bandlimited signal having a frequency spectrum with non-zeroFourier coefficients for frequencies higher than half of the samplingfrequency 2ω_(m), current sampling and reconstruction methods imply adistortion of the reconstructed signal with respect to the sampledsignal.

In order to avoid the above mentioned problem, one usually uses a highersampling frequency, thus allowing perfect reconstruction of a broaderrange of signals and minimizing distortion for non-bandlimited signalsor wideband signals. A high sampling frequency however requires fast,expensive and power-consuming A/D converters, fast digital circuits anda waste of storage place for storing the digitized signal.

Furthermore, there are wide classes of very common signals, includingstream of Dirac pulses, bilevel signals, piecewise polynomial signals,etc., which are not bandlimited and which therefore cannot be sampledand faithfully reconstructed by the known methods, even by increasingthe sampling rate.

An aim of the present invention is to find a sampling method and arelated reconstruction method for sampling at least some classes ofnon-bandlimited signals and for allowing an exact reconstruction ofthese signals from the samples generated with said sampling method.

Another aim of the present invention is to find a sampling method and arelated reconstruction method for sampling at least some classes ofbandlimited signals with a sampling frequency lower than the frequencygiven by the Shannon theorem and still allowing an exact reconstructionof these signals.

Another aim of the present invention is to find an improved method forsampling and faithfully reconstructing signals with a finite rate ofinnovation which can only be seen through an imperfect measuring device,a transfer channel and/or a modulating system having not necessarily abandlimited transfer characteristic.

These aims are achieved with a sampling method and a reconstructionmethod including the features of the corresponding independent claim.

In particular, these aims are achieved with a new sampling method forsampling and faithfully reconstructing signals having a finite rate ofinnovation, i.e. having, over a finite time interval, a finite number 2Kof degrees of freedom. In particular, the invention concerns thesampling and reconstruction of signals which can be completelyrepresented by the superposition of a finite number K of known functionsdelayed by arbitrary shifts (t_(n), t_(k)) and weighted by arbitraryamplitude coefficients c_(n), c_(k).

The Shannon theorem indicates that bandwidth limited signals can beexactly specified by sampling values y_(s)[n] taken at uniform samplingintervals if the sampling rate is at least twice the bandwidth. Theinvention is based on the finding that a larger class of signals, i.e.signals having a finite number 2K of degrees of freedom over a finitetime interval, can be exactly specified by a finite set of shifts andassociated weights. It can be shown that, in many common cases, theminimal number of values 2K, or even 2K+1, is much lower than the numberof samples required for faithfully reconstructing a signal with theexisting reconstruction methods.

As an example, in the case of a CDMA communication system, each symbol(information bit) of each signal sent by each user is modulated with acoding sequence (signature) which can for example be 1023 chips long. Inthis case, the chip rate of the transmitted signal is thus 1023 timeshigher than its symbol rate. Modulating a signal with the codingsequence thus expands the signal's bandwidth by the value of thespreading factor, here 1023. Sampling and reconstructing a received CDMAsignal with the conventional sampling and reconstruction methods thusrequires a very fast and therefore complex and expensive analog samplingdevice. As the number of degrees of freedom in each received signatureis at most two (shift and weight of the signature), the method of theinvention can be used for sampling the received signal at a much lowerrate whilst still allowing a faithful reconstruction of the sequence ofsymbols sent.

It is important to understand that, in the sampling method of theinvention, the signal (or a filtered version of the signal) is stillsampled at uniform sampling intervals, using for example commonregularly clocked sampling devices. It is only for the reconstruction ofthe signal from the set of sampled values that a set of shifted valuesis computed. In most cases, those shifted values do not correspond tosamples of the signal to reconstruct.

The inventive sampling method first convolves the signal x(t) with asampling kernel and then samples the convolved signal at regularsampling intervals, both the sampling kernel and the sampling frequencybeing chosen such that the sampled values completely specify the firstsignal, thus allowing a perfect reconstruction of said first signal. Thesampling frequency used for the inventive sampling method can be lowerthan the frequency given by the Shannon theorem, but is greater than orequal to the rate of innovation of the signal to reconstruct.

The inventive reconstruction method reconstructs a first signal x(t)from a set of sampled values taken from this signal x(t) or from asecond related signal y(t) regularly sampled at a sub-Nyquist rate byretrieving from the regularly spaced sampled values a set of shifts andweights with which the first signal can be completely specified andreconstructed.

In particular, the inventive reconstruction method can be used forreconstructing, from a set of at least 2K sampled values, any signalwhich can be represented by the superposition of K known functionsdelayed by arbitrary shifts and weighted by arbitrary amplitudecoefficients (weights). A preferred embodiment of the inventivereconstruction method comprises the steps of first solving a structuredlinear system for retrieving said arbitrary shifts and then retrievingthe arbitrary weights using the previously retrieved arbitrary shifts.

Note that for some applications it may be sufficient to retrieve theshifts and that the weights are only needed if a complete reconstructionof the signal is needed. This is for instance the case during anestimation session in a CDMA decoder, when one wants to estimate therelative delays (shifts) occurred by the different users' signals alongdifferent propagation paths.

The lowest sampling frequency required in the method of the inventiondirectly depends on the rate of innovation of the signal to sampleand/or reconstruct, and not on its bandwidth as in prior art methods.More precisely, the sampling frequency required by the sampling methodof the invention must be greater than or equal to the rate of innovationof the signal to sample. A minimal sampling frequency can thus bedetermined for any signal with a finite rate of innovation, includingfor some non-bandlimited signals.

In the specification and in the claims, the rate of innovation ρ of asignal is defined as the number of degrees of freedom of the signalwithin a time period. For example, a signal x(t) made of K weightedDirac pulses, even if clearly not bandlimited, can be fully specified bythe K occurrence times (shifts) t_(k) and by the K amplitudes (weights)c_(k) of the Diracs. The signal can be written as

${x(t)} = {\sum\limits_{k = 0}^{K - 1}{c_{k}{{\delta \left( {t - t_{k}} \right)}.}}}$

The degree of freedom of this signal is 2K. Its rate of innovation ρ isthus finite and equal to 2K/τ, where τ is the time interval in whichthese K Dirac pulses occurred.

If the Dirac pulse function δ(t) is replaced by any other known functionγ(t), the number of degrees of freedom of the signal obviously alwaysstays 2K, that is K occurrence times, or shifts, t_(k) and K amplitudes,or weights, c_(k).

Considering the more general case of an unlimited time-continuoussignal, for example a sequence of regularly spaced Dirac pulses with aspace T between the pulses, the signal can be represented as

${x(t)} = {\sum\limits_{n = {- \infty}}^{\infty}{c_{n}{\delta \left( \frac{t - {nT}}{T} \right)}}}$

where the degrees of freedom of the signal are the coefficients c_(n).The number of degrees of freedom per time interval T is therefore equalto one. The rate of innovation of this signal is thus ρ=1/T. Such asignal could for example be a Pulse Amplitude Modulated (PAM) signalwhere the coefficients c_(n) represent the value of the information datato be transmitted.

Again, replacing the Dirac pulse function δ(t) by any other knownfunction γ(t) doesn't change the rate of innovation of the signal. Thesignal

${x(t)} = {\sum\limits_{n = {- \infty}}^{\infty}{c_{n}{\gamma \left( \frac{t - {nT}}{T} \right)}}}$

thus also has a rate of innovation ρ=1/T. There are many examples ofsuch signals, for example when γ(t) is a scaling function in a waveletmulti-resolution framework, or in approximation theory using uniformsplines.

If the superposed copies of the function y(t) are shifted by arbitraryshifts t_(n), the signal can be written as

${x(t)} = {\sum\limits_{n = {- \infty}}^{\infty}{c_{n}{\gamma \left( \frac{t - t_{n}}{T} \right)}}}$

Assuming that the function y(t) is known, the degrees of freedom of thesignals over a time interval T are the weights c_(n) and the shiftst_(n). Thus the rate of innovation is ρ=2π/T.

A signal could also be represented as a superposition of a determinedset of functions {γ_(r)(t)}_(r=0, . . . , R), instead of the uniquefunction γ(t). It can thus be written as

${x(t)} = {\sum\limits_{n = {- \infty}}^{\infty}{\sum\limits_{r = 0}^{R}{c_{nr}{\gamma_{r}\left( \frac{t - t_{n}}{T} \right)}}}}$

For example, γ_(r) could be a Dirac, a differentiated Dirac, apolynomial function, etc. Likewise, a multi-user CDMA signal x(t) is thesum of a plurality of weighted and shifted known signatures γ_(r) of theusers.

Again, assuming that the functions γ_(r)(t) are known, the degrees offreedom of the signal x(t) are the coefficients c_(nr) and the timeinstants t_(n). With the introduction of a counting functionc_(x)(t_(a),t_(b)) which counts the number of degrees of freedom in thesignal x(t) over a time interval [t_(a),t_(b)], the rate of innovation ρcan be defined as

$\rho = {\lim\limits_{\tau->\infty}{\frac{1}{\tau}{c_{x}\left( {{- \frac{\tau}{2}},\frac{\tau}{2}} \right)}}}$

Signals with a finite rate of innovation ρ can be completely determinedand faithfully reconstructed with a set of at least ρ/2 weights (c_(n),c_(nr), c_(k)) and ρ/2 shifts (t_(n), t_(k)) per unit of time and withthe knowledge of the functions γ_(r)(t) which only depend on the classof the signal to reconstruct (stream of pulses, piecewise polynomials,etc.).

The rate of innovation ρ can also be defined locally with respect to amoving window of size τ. Given a window of time τ, the local rate ofinnovation ρ_(r)(t) at time t is

${\rho_{t}(t)} = {\frac{1}{\tau}{{C_{x}\left( {{t - {\tau/2}},{t + {\tau/2}}} \right)}.}}$

In this case, as the rate of innovation ρ_(r) of the signal determinesthe lowest required sampling frequency needed for sampling the signalwith the inventive sampling method, it is necessary to determine themaximal local rate of innovation ρ_(max)(τ)

ρ_(max)maxρ_(τ)(t).

Applications of the new sampling and reconstruction methods can be foundin many technical fields, including signal processing, communicationsystems and biological systems. The sampling and reconstruction methodscan for instance be used for sampling and reconstructing widebandcommunication signals, such as CDMA signals, or ultra-widebandcommunication signals, such as PPM signals and time-modulatedultra-wideband signals, among others.

Various embodiments of the new sampling method and of the relatedreconstruction method applied to different classes of signals withfinite rate of innovation are explained in the specification. The oneskilled in the art will understand however that the invention is notlimited to the specific examples given and that advantageous technicaleffects can be obtained by sampling and reconstructing other kinds ofsignals with a finite rate of innovation.

The invention will be better understood with the help of the figures inwhich:

FIG. 1 diagrammatically illustrates a sampling device and method usingthe already known Shannon theorem.

FIG. 2 diagrammatically illustrates a reconstructing device and methodusing the already known Shannon theorem.

FIG. 3 illustrates a communication system in which the sampling andreconstruction method of the invention can be used for sampling thesignal y(t) and reconstructing any of the signals x_(i)(t).

FIG. 4 illustrates a sampling device according to the invention.

FIG. 5 shows one period τ of a periodic stream of Diracs x(t).

FIG. 6 shows the Fourier transformation of a periodic stream of Diracs.

FIG. 7 shows the Fourier transformation of a bandlimited periodic streamof Diracs.

FIG. 8 shows the Fourier transformation of a sampled bandlimited,periodic stream of Diracs.

FIG. 9 shows an example of bilevel signal.

FIG. 10 shows a box spline φ₀(t/T).

FIG. 11 shows a hat spline φ₁(t/T).

FIG. 12 illustrates a two-dimensional picture signal with a 0/1transition given by an arbitrary function.

FIG. 13 illustrates a two-dimensional picture signal with a 0/1transition given by a smooth, for example a bandlimited, function.

FIG. 14 illustrates a two-dimensional picture signal with a 0/1transition given by a piecewise polynomial function.

FIG. 3 diagrammatically illustrates a communication system in which thesampling and reconstruction method of the invention can be used. Thesystem comprises a signal x_(τ)(t) which is modulated by a modulatingdevice 3 having a known transfer function φ₁(t). The resulting modulatedsignal x₂(t) is transmitted over a transmission channel 5 with atransfer function φ₂(t). Noise may be added to the signal x₃(t). Thetransmitted signal x₃(t) is acquired or measured by a measuring device 7with a transfer function φ₃(t). The measured signal x₄(t) is demodulatedby a demodulating device 9 with a known transfer function φ₄(t). Thedemodulated signal x₅(t) is filtered by a filter 11 with a knowntransfer function φ₅(t), for instance a lowpass filter in the sampler.In the following, the combination of transfer functions φ_(i)(t) bywhich the sampled signal y(t) is related to the signal x_(i)(t) onewishes to reconstruct will be designated by φ(t), and the reconstructedsignal will be designated x(t). The filtered signal x₆(t)=y(t) issampled at uniform sampling intervals by a sampling device 13 working ata sub-Nyquist rate, generating a set of sampled values y_(s)[n]. Thisset of sampled values can be stored, processed, transmitted, etc., andused by a reconstruction device 15 for reconstructing the signal y(t)or, if the transfer functions φ_(i)(t) are known or at least if theyhave themselves a finite rate of innovation, any of the signalsx_(i)(t). Depending on the application, one may wish to reconstruct thesampled signal y(t) or any signal x_(i)(t) related to the sampled signaly(t) by a known transfer function, or at least by a transfer functionwhich has itself a finite rate of innovation.

FIG. 4 illustrates a sampling device 13 with a filter 11 able to carryout the sampling method of the invention. The aim of the sampling deviceis to generate a set of sampled values y_(s)[n] from the signal y(t)having a known finite rate of innovation ρ, where this set of valuey_(s)[n] must completely specify the signal to reconstruct x(t).

According to the sampling method of the invention, the signal to samplex(t) is first filtered with a lowpass filter, for example with thelowpass filter 11 or with a combination of the elements 3-11 in thesystem of FIG. 3, having an impulse response φ(t), for instanceφ_(s)(t). Advantageously, the impulse response φ_(s)(t) of the lowpassfilter has a bandwidth of ρ/2.

Calling x(t) the signal to reconstruct, its filtered version is

y(t)=x(t)*{tilde over (φ)}(t)

where

{tilde over (φ)}(t)=φ(−t)

is the convolution kernel (transfer function of the filter 3-11). Then,uniform sampling of y(t) with a sampling interval T leads to samplesy_(s)[n] given by:

y_(s)[n] = ⟨ϕ(t − nT), x(t)⟩ = ∫_(−∞)^(∞)ϕ(t − nT)x(t)t

By choosing a sampling frequency f=1/T greater or equal to the rate ofinnovation of the signal x(t) to reconstruct, we will show that thesamples y_(s)[n] allow for a faithful reconstruction or decoding of thesignal x(t), as will be illustrated by the not-limiting followingexamples.

Periodic Signals with Finite Rate of Innovation

In an embodiment, the new sampling and reconstruction methods areapplied to the sampling and/or reconstruction of a periodic first signalx(t) with finite rate of innovation ρ over its period τ. As examples, wewill consider streams of weighted Diracs and periodic piecewisepolynomials. Although the demonstration will be made usingcontinuous-time periodic signals, similar results could be achieved withdiscrete-time periodic signals.

A periodic stream of K Diracs at locations t_(k), weighted by weightsc_(k) and with a period τ, can be written as

${x(t)} = {\sum\limits_{n}{\sum\limits_{k = 0}^{K - 1}{c_{k}{\delta \left( {t - \left( {t_{k} + {n\; \tau}} \right)} \right)}}}}$

One period τ of a stream of Diracs x(t) is shown on FIG. 5. Each periodof x(t) can thus be represented by the superposition of K Diracs delayedby shifts t_(k) and weighted by weights c_(k). This signal x(t) can thusbe fully determined by knowing the values of the K amplitudes and the Kshifts of the K Diracs. It has therefore 2K degrees of freedom perperiod τ and its rate of innovation ρ is equal to 2K/t. However, thebandwidth of x(t) is clearly not limited, so that x(t) cannot be sampledand faithfully reconstructed with existing methods.

According to the sampling method of the invention, this first signalx(t) is first convolved by a filter 3-11 with a sampling function φ(t),for instance the sin c sampling kernel of bandwidth

$\left\lbrack {\frac{- K}{\tau},\frac{K}{\tau}} \right\rbrack,$

and then sampled at a sampling frequency f=1/T greater than

$\frac{2K}{\tau}.$

The Fourier series coefficients X[m] of the signal x(t) are given by

$\begin{matrix}{{X\lbrack m\rbrack} = {\frac{1}{\tau}{\sum\limits_{k = 0}^{K - 1}{c_{k}^{\; \frac{2\; \pi \; m\; t}{\tau}}}}}} & \left( {{Figure}\mspace{14mu} 6} \right)\end{matrix}$

The filtered signal y(t), resulting from the convolution of the firstsignal x(t) with the sin c sampling kernel of bandwidth

$\left\lbrack {\frac{- K}{\tau},\frac{K}{\tau}} \right\rbrack$

is a lowpass approximation of the first signal x(t) given by

${y(t)} = {\sum\limits_{m = {- k}}^{K}{{X\lbrack m\rbrack}^{\; \frac{2\pi \; {mt}}{\tau}}}}$

FIG. 7 shows the Fourier series Y[m] coefficients of y(t).

Sampling one period τ of the filtered signal y(t) at multiples of T, weobtain M=τ/T sampled values y_(s)[n]. In a preferred embodiment, T is adivider of τ.

${{y_{s}\lbrack n\rbrack} = {{y({nT})} = {{\sum\limits_{m = {- K}}^{K}{{X\lbrack m\rbrack}^{\; \frac{2\pi \; m\; t}{\tau}}}} = {< {x(t)}}}}},{{{\phi \left( {t - {nT}} \right)} > {{where}\mspace{14mu} n}} \in {\left\lbrack {0,{{\tau/T} - 1}} \right\rbrack.}}$

y(nT) is periodic with a period τ/T. FIG. 8 shows the Fourier seriescoefficients Y_(s)[m] of y_(s)[n]. The Fourier coefficients Y_(s)[m] canbe computed from the set of N sampled values y(nT) of the filteredsignal y(t) by using for instance the well-known Fast Fourier Transform(FFT) method.

It can be shown that if the number τ/T of samples per period is greaterthan or equal to 2K+1, the samples y_(s)[n] are a sufficientrepresentation of x(t).

The following method can be used for reconstructing the signal x(t) fromthe coefficients Y_(s)[m]. The Fourier coefficients X[m], computed fromthe samples y_(s)[n] with a known method, are a linear combination of Kcomplex exponentials:

$u_{k}^{m} = ^{{- }\; \frac{2\pi \; {mt}_{k}}{\tau}}$

The shifts t_(k) of the K Diracs can be determined from the samplesusing for instance an annihilating filter. A filter (1−z⁻¹u_(k)) iscalled an annihilating filter for u_(k) ^(m) if

(1−z ⁻¹ u _(k))u _(k) ^(m)=0

In order to find the shifts t_(k), an annihilating filter H(z) has to bedetermined whose coefficients are (1, H[1], H[2], . . . , H[K]) or

${H(z)} = {{1 + {{H\lbrack 1\rbrack}z^{- 1}} + {{H\lbrack 2\rbrack}z^{- 2}} + \ldots + {{H\lbrack K\rbrack}z^{- K}}} = {\sum\limits_{m = 0}^{K}{{H\lbrack m\rbrack}z^{- m}}}}$

and which annihilates each exponential u_(k) ^(m). It can be shown thatthe K shifts of the Diracs {t₀, t₁, . . . , t_(k−1)} are given by, or atleast can be retrieved from the zeros of the filter H(z).

The filter H(z) can be found by solving the following structured linearequation system for H:

${\begin{bmatrix}{X\lbrack 0\rbrack} & {X\left\lbrack {- 1} \right\rbrack} & \ldots & {X\left\lbrack {- \left( {K - 1} \right)} \right\rbrack} \\{X\lbrack 1\rbrack} & {X\lbrack 0\rbrack} & \ldots & {X\left\lbrack {- \left( {K - 2} \right)} \right\rbrack} \\\vdots & \; & \; & \vdots \\{X\left\lbrack {K - 1} \right\rbrack} & {X\left\lbrack {K - 2} \right\rbrack} & \ldots & {X\lbrack 0\rbrack}\end{bmatrix} \cdot \begin{bmatrix}{H\lbrack 1\rbrack} \\{H\lbrack 2\rbrack} \\\vdots \\{H\lbrack K\rbrack}\end{bmatrix}} = {- \begin{bmatrix}{X\lbrack 1\rbrack} \\{X\lbrack 2\rbrack} \\\vdots \\{X\lbrack K\rbrack}\end{bmatrix}}$

Because the matrix is a Toeplitz system, fast algorithms are availablefor finding the solution. This system has a unique solution if the firstmatrix is invertible, which is the case if all K Diracs in x(t) aredistinct, that is if t_(k)≠t_(l), ∀k≠l.

Given the coefficients 1, H[1], H[2], . . . , H[K] the filter H(z) canbe factored into its roots

${H(z)} = {\prod\limits_{k = 0}^{K - 1}\left( {1 - {u_{k}z^{- 1}}} \right)}$

which leads to the K shifts t_(k) using the above given relation betweenu_(k) and t_(k).

Given the shifts t_(k), the K values of the weights c_(k) can be foundby solving

${X\lbrack m\rbrack} = {\frac{1}{\tau}{\sum\limits_{k = 0}^{K - 1}{c_{k}^{{{- }\; \frac{2\pi \; m\; t_{k}}{\tau}}\;}}}}$

which leads to the following Vandermonde system

$\begin{bmatrix}{X\lbrack 0\rbrack} \\{X\lbrack 1\rbrack} \\\vdots \\{X\left\lbrack {K - 1} \right\rbrack}\end{bmatrix} = {{\frac{1}{\tau}\begin{bmatrix}1 & 1 & \ldots & 1 \\u_{0} & u_{1} & \ldots & u_{K - 1} \\\; & \; & \; & \vdots \\u_{0}^{K - 1} & u_{1}^{K - 1} & \ldots & u_{K - 1}^{K - 1}\end{bmatrix}} \cdot \begin{bmatrix}c_{0} \\c_{1} \\\vdots \\c_{K - 1}\end{bmatrix}}$

which again has a unique solution when the K shifts t_(k) are distinct,t_(k)≠t_(l), ∀k≠l. Because the matrix is a K×K Vandermonde equationsystem, known fast solution methods are available.

This result can easily be extended to a periodic stream ofdifferentiated Diracs:

${x(t)} = {\sum\limits_{n = {- \infty}}^{\infty}{\sum\limits_{n = 0}^{R_{n} - 1}{c_{nr}{\delta^{(r)}\left( {t - t_{n}} \right)}}}}$

with the periodicity conditions t_(n+K)=t_(n)+τ and c_(n+K,r)=c_(nr) forall n.

This first signal x(t) is entirely determined by the K shifts t_(k) andthe K′ weights c_(nr), where

$K^{\prime} = {\sum\limits_{k = 0}^{K - 1}{R_{k}.}}$

which makes at most K+K′ degrees of freedom per period τ. The rate ofinnovation ρ is thus

$\rho = {\frac{K + K^{\prime}}{\tau}.}$

Applying the new sampling method as in the previous examples, thissignal is first convolved with a sampling kernel of a bandwidth Bgreater than or equal to the rate of innovation ρ of said first signalx(t), then sampled at a frequency greater than or equal to the maximumbetween B and ρ. Note that even sampling at a frequency ρ when thebandwidth B is greater than the rate of innovation ρ leads to a limitednumber of solutions among which the good solution may often be guessed.

The so generated sampled values y(nT) will then be sufficient to recoverthe spectral values X[m] using similar steps of the variant embodimentof the reconstruction method used in the case where x(t) is a periodicstream of weighted Diracs, leading then similarly to the recovery of theshifts t_(k) and the K′ weights c_(nr).

We will now extend this result to signals x(t) belonging to the class ofperiodic piecewise polynomial signals of period τ, containing in eachperiod K pieces of maximum degree R. If we differentiate a periodicpiecewise polynomial x(t) R+1 times, we obtain a stream of K weightedand shifted Diracs or derivative of Diracs:

${X^{({R + 1})}(t)} = {\sum\limits_{n = {- \infty}}^{\infty}{\sum\limits_{r = 0}^{R_{n} - 1}{c_{nr}{\delta^{(r)}\left( {t - t_{n}} \right)}}}}$

Periodic piecewise polynomials signals x(t) are thus entirely determinedby K shifts t_(k) and K′=(R+1)K weights. The rate of innovation ρ is

$\rho = {\frac{\left( {R + 2} \right)K}{\tau}.}$

The Fourier coefficients of the derivative operator is defined byD[m]=i2πm and therefore the Fourier coefficients X^((R+1))[m] of thedifferentiated signal x^((R+1))(t) are equal to

X ^((R+1)) [m]=(i2πm/τ)^(R+1) X[m]

with the periodicity conditions t_(n+K)=t_(n)+τ and c_(n+K,r)=c_(nr) forall n.

This first signal x(t) is entirely determined by the K shifts t_(k) andthe K′=(R+1)K weights. The rate of innovation ρ is thus

$\rho = {\frac{\left( {R + 2} \right)K}{\tau}.}$

Applying the new sampling method as in the previous example, x(t) isfirst convolved with a kernel φ(t) of a bandwidth B greater than orequal to the rate of innovation ρ of said first signal x(t), for examplewith the differentiated sin c sampling kernel, or with anotherdifferentiated kernel.

We can then take τ/T samples regularly spaced apart in order to get aset of values y_(s)[n] from which the periodic piecewise polynomialsignal of degree R x(t) can be faithfully reconstituted using thepreviously described annihilating filter method.

A similar reconstruction method can be applied for sampling andreconstructing periodic non-uniform splines of degree R, i.e. signalswhose (R+1)th derivative is a periodic stream of K weighted and shiftedDiracs. Likewise, a similar reconstruction method can be applied forperiodic filtered piecewise polynomial signals.

Furthermore, this sampling and reconstruction method can be applied topiecewise bandlimited signals, i.e. to signals which are bandlimitedexcept for jumps or discontinuities. Piecewise bandlimited signals canbe seen as the sum of bandlimited signals with a stream of Diracs and/orwith a piecewise polynomial signal:

x(t)=x_(BL)(t)+x_(PP)(t) where x_(BL)(t) is a L-Bandlimited signal andx_(PP)(t) is a piecewise polynomial signal.

The Fourier coefficients X[m] are defined by

${X\lbrack m\rbrack}\left\{ \begin{matrix}{{X_{BL}\lbrack m\rbrack} + {X_{pp}\lbrack m\rbrack}} & {{{if}\mspace{14mu} m} \in \left\lbrack {{- L},L} \right\rbrack} \\{X_{pp}\lbrack m\rbrack} & {{{if}\mspace{14mu} m} \notin \left\lbrack {{- L},L} \right\rbrack}\end{matrix} \right.$

The Fourier coefficients of the signal x(t) outside of the band [−L,L]are exactly equal to the Fourier coefficients of the piecewisepolynomial. Therefore it is sufficient to take at least 2K(R+1) Fouriercoefficients outside of the band [−L,L] to retrieve the signalx_(PP)(t). The Fourier coefficients of the bandlimited signal are thenobtained by subtracting X_(PP)[m] from X[m] for mε[−L,L].

As a piecewise polynomial signal has 2K(R+1) and the bandlimited signal2L+1 degrees of freedom, we can sample the signal x(t) using aperiodized differentiated sin c sampling kernel bandlimited to2L+4K(R+1)+1, because of symmetry constraints.

Finite Length Signals with Finite Rate of Innovation

A finite length signal with finite rate of innovation p clearly has afinite number of degrees of freedom. We will illustrate with severalexamples that those signals can be uniquely specified with a finite setof samples, and that the minimal number of samples in the set onlydepends on the number of degrees of freedom of the signal, but not onits bandwidth.

Consider first a continuous time signal x(t) with a finite number ofweighted Diracs:

${x(t)} = {\sum\limits_{k = 0}^{K - 1}\; {c_{k}\; {\delta \left( {t - t_{k}} \right)}}}$

x(t) clearly has 2K degrees of freedom, K from the weights c_(k) and Kfrom the shifts t_(k) of the Diracs. We will see that N samples, N beinggreater than 2K, preferably greater than 2K+1, will be sufficient torecover the signal x(t), and describe an appropriate reconstructionmethod. Similar to the previous cases, the reconstruction method willrequire solving two systems of linear equations: one for the shifts ofthe Diracs and one for the weights of the Diracs.

In a first embodiment, the signal x(t) is filtered with a sin c kernelφ(t), as an example of infinite length sampling kernel. Sampled valuesy_(s)[n] of x(t) are obtained by filtering x(t) with the sin c(t/T)sampling kernel:

y _(s) [n]=<x(t),sin c(t/T−n)>, n=0, . . . , N−1

By developing the inner product one shows that

${y_{s}\lbrack n\rbrack} = {\frac{\left( {- 1} \right)^{n}}{\pi}{\sum\limits_{k = 0}^{K - 1}{c_{k}{{\sin \left( {\text{?}_{k}{IT}} \right)} \cdot \frac{1}{\left( {{t_{k}{IT}} - n} \right)}}}}}$?indicates text missing or illegible when filed

This leads to:

$Y_{n} = {{\left( {- 1} \right)^{n}{P(n)}{y_{s}\lbrack n\rbrack}} = {\frac{1}{\pi}{\sum\limits_{k = 0}^{K - 1}{c_{k}{\sin \left( {\text{?}_{k}{IT}} \right)}{P_{k}(n)}}}}}$${{where}\mspace{14mu} {P(u)}} = {\sum\limits_{k = 0}^{K}{p_{k}u^{k}}}$?indicates text missing or illegible when filed

has zeros at locations t_(l)/T for l=0, . . . , K−1

The right-hand side of this expression is a polynomial of degree K−1 inthe variable n, thus applying K finite differences makes the left-handside vanish, that is

${{\Delta^{K}\left( {\left( {- 1} \right)^{n}{P\lbrack n\rbrack}{y_{s}\lbrack n\rbrack}} \right)} = {{0\mspace{14mu} {for}\mspace{14mu} n} = K}},\ldots \mspace{14mu},{\left. {N - 1}\Leftrightarrow{\sum\limits_{k = 0}^{K}\; {p_{k}{\Delta^{k}\left( {\left( {- 1} \right)^{n}n^{k}{y_{s}\lbrack n\rbrack}} \right)}}} \right. = 0}$

The p_(k) can be retrieved by solving this system. With the p_(k), the Kroots of the polynomial P(u) can be retrieved, and thus the shiftst_(k)=Tu_(k). Once the shifts t_(k) have been retrieved, the weightsc_(k) can be found with the above described method, i.e. by solving alinear system of equations.

The above method for reconstructing a signal which has been filtered bya sin c transfer function is easy to carry out and elegant. However, inthe real world, many signals are measured by a measuring device 7respectively transmitted over a channel 5 having a transfer functionwhich can be more precisely approximated by a Gaussian transfer functionφ_(σ)(t). It can be shown that similar to the sin c sampling kernel, 2Ksample values y_(s)[n] obtained by filtering the signal with a Gaussiankernel φ_(σ)(t) are sufficient to represent the signal x(t).

This can be demonstrated by computing the sample values

${{y_{s}\lbrack n\rbrack} = {< {x(t)}}},{^{{- {({{t/T} - n})}}\text{?}}>={\sum\limits_{k = 0}^{K - 1}{c_{k}^{{- {({{\text{?}{IT}} - n})}}\text{?}}}}}$?indicates text missing or illegible when filed?

By expanding and regrouping the terms so as to have variables thatdepend solely on n and solely on k, we obtain:

$Y_{n}{\sum\limits_{k = 0}^{K - 1}\; {a_{k}u_{k}^{n}}}$

where we have Y_(n)=e^(n) ² ^(/2σ) ² y_(x)[n], a_(k)=c_(k)e^(−r) ^(k) ²^(/2σ) ² ^(T) ² and u_(k)=e^(r) ^(k) ^(/σ) ² ^(T). Y_(n) is a linearcombination of real exponentials. Thus the annihilating filter methodcan be used to find the K values u_(k) and a_(k). This means that thevalues u_(k) can be solved by finding the roots of an annihilatingfilter chosen such that

${{h*Y} = {\left. 0^{\prime}\Leftrightarrow{\sum\limits_{k = 0}^{K}\; {h_{k}Y_{n - k}}} \right. = 0}},{n = K},\ldots \mspace{14mu},{N - 1}$

From the u_(k), the shifts t_(k) can be retrieved using t_(k)=σ²T lnu_(k). Once the shifts t_(k) are obtained then we solve for a_(k) aVandermonde equation system. The weights c_(k) are simply given byc_(k)=a_(k)e^(r) ^(k) ² ^(/2σ) ² ^(T) ²

Infinite Length Signals with Local Finite Rate of Innovation

In this section, we will describe a sampling and local reconstructionmethod for sampling and reconstructing infinite length signals x(t) witha finite local rate of innovation ρ_(τ)(t) with respect to a movingwindow of size t. We will in particular describe bilevel signals, andexplain local reconstruction methods using sampling kernels with acompact support. Those results could be generalized to other B-splinesof different degree d.

Bilevel signals x(t) are infinite length continuous-time signals whichtake on two values, 0 and 1, with a known initial condition, for examplex(0)=1. These signals are completely represented by their transitionvalues (shifts) t_(k). Suppose the signal x(t) has a finite local rateof innovation ρ as described above. This is the case of most signalsproduced by electronic digital circuits, where the rate of innovation isusually limited by the clocking of the integrated circuit generating thesignal. Examples of common bilevel signals with a finite local rate ofinnovation include amplitude or modulation pulses or PAM, PPM signalsamong others. An example of bilevel signal is shown on FIG. 9.

If a bilevel signal is sampled with a box spline φ₀(t/T) shown on FIG.10, then the sample values y_(s)[n] are given by the inner productbetween the bilevel signal and the box function:

y_(s)[n] =  < x(t), ϕ₀(t/T − n) > ∫_(−∞)^(∞)x(t)ϕ₀(t/T − n) t

The sample value y_(s)[n] simply corresponds to the area occupied by thesignal x(t) in the interval [nT, (n+1)T]. Thus, if there is at most onetransition per box, then we can recover the transition from the sample.The non-bandwidth limited signal x(t) is thus uniquely determined fromthe finite set of samples y_(s)[n].

However, if the bilevel signal x(t) is shifted by an unknown shift, thenthere may be two transitions in an interval of length T and one boxfunction will not be sufficient to recover the transitions. To samplethis signal, one can use a sampling kernel φ(t/T) with a larger supportand with added information. For example, the hat spline function definedby

$\begin{matrix}{{\phi_{1}(t)} = \left\{ \begin{matrix}{1 - {t}} & {{{if}\mspace{14mu} {t}} < 1} \\0 & {else}\end{matrix} \right.} & \left( {{Fig}.\mspace{14mu} 11} \right)\end{matrix}$

leads to two sample values in each interval [nT, (n+1)T] with which themaximum two transitions time t_(k) (shifts) in the intervals can beuniquely retrieved.

In fact, if there are at most 2 transitions in the interval [n,n+2],then the possible configurations are

(0,0), (0,1), (0,2), (1,0), (1,1), (2,0)

where the first and second component indicate the number of transitionsin the intervals [n, n+1] and [n+1, n+2] respectively.

Since the hat sampling kernel is of degree one, we obtain for eachconfiguration a quadratic system of equations with variables t₀, t₁:

     y_(n) = ∫_(n − 1)^(n)x(t)(1 + t − n) t + ∫_(n)^(n + 1)x(t)(1 − (t − n)) t   y_(n + 1) = ∫_(n)^(n + 1)x(t)(1 + t − (n + 1)) t + ∫_(n + 1)^(n + 2)x(t)(1 − (t − (n + 1))) t

It can easily be shown that this system of equation admits one solutionand that this solution is unique.

Similarly, infinite length piecewise polynomial signals with a localfinite rate of innovation can be sampled and reconstructed with a boxsampling kernel. Consider an infinite length piecewise polynomial signalx(t) where each piece is a polynomial of degree R and defined over aninterval [t_(k−1),t_(k)], that is,

${x(t)}\left\{ \begin{matrix}{{x_{0}(t)} = {\sum\limits_{m = 0}^{R}\; {c_{0m}t^{m}}}} & {t \in \left\lbrack {0,t_{0}} \right\rbrack} \\{{x_{1}(t)} = {\sum\limits_{m = 0}^{R}\; {c_{1m}t^{m}}}} & {t \in \left\lbrack {t_{0},t_{1}} \right\rbrack} \\\ldots & \; \\{{x_{K}(t)} = {\sum\limits_{m = 0}^{R}\; {c_{Km}t}}} & {t \in \left\lbrack {t_{K - 1},t_{K}} \right\rbrack} \\\ldots & \;\end{matrix} \right.$

Each polynomial piece x_(k)(t) contains R+1 unknown coefficients c_(km).The transition values are easily obtained once the pieces x_(k−1)(t) andx_(k)(t) are determined, thus there are 2(R+1)+1 degrees of freedom. Ifthere is one transition in an interval of length T then the maximallocal rate of innovation is ρ_(m)(T)=(2(R+1)+1)/T. Therefore, in orderto recover the polynomial pieces and the transition we need to have atleast 2(R+1)+1 samples per interval T. This can be achieved for exampleby sampling with the following box sampling kernel:

$\phi_{0}\left( {{tl}\frac{T}{{2\left( {R + 1} \right)} + 1}} \right)$

For example, if the signal x(t) to reconstruct is a piecewise linearsignal (R=1) with maximal one transition in each interval T, then weneed at least 5 samples per interval.

We can generalize this case by noting that the Rth derivative of apiecewise polynomial of degree R is a piecewise constant signal whichcan be sampled and reconstructed using the same method.

Multidimensional Signals

The sampling and reconstruction method of the invention is notrestricted to one-dimensional signals, but can also be used withmultidimensional signals. For example, an interesting case appears withtwo-dimensional images, where bilevel and multi-level signals are quitecommon. Furthermore, many two-dimensional picture signals have a finitelength and can only be seen through an optical system with a transferfunction which is at least approximately Gaussian.

Consider the cases shown on FIGS. 12, 13 and 14:

FIG. 12: unit square with a 0/1 transition given by an arbitraryfunction.

FIG. 13: unit square, with a 0/1 transition given by a smooth, forexample bandlimited, function.

FIG. 14: unit square, with a 0/1 transition given by a piecewisepolynomial function.

The sampling and reconstruction method developed in the previoussections can be applied on a set of lines, for example on a square grid,through the square. Obviously, a perfect reconstruction of theboundaries shown on FIGS. 13 and 14 is possible, but a priori not forthe case illustrated on FIG. 12. Depending on the a-priori known classof function to which the boundary function belongs, a different samplingkernel φ(t) will be used for sampling lines through the image and toallow a perfect reconstruction of the image if the sampling is fineenough. For instance, if the image is piecewise polynomial withboundaries that are either bandlimited or piecewise polynomial, thenseparable one-dimensional sampling using spline kernels is possible thatallows, with the above described method, a perfect reconstruction of theboundary and thus of the bilevel image, if the sampling is fine enough,i.e. if the sampling rate is higher than the rate of innovation of theboundary function.

Instead of scanning a picture along lines with a one-dimensionalsampling kernel, one can scan it with a two-dimensional sampling kernel.Indeed, this is exactly what a scanner or a digital camera does: theimage is first filtered by the optical system which has atwo-dimensional φ(x,y) transfer function and then sampled at regularintervals by the matrix sensor. In fact, the above-described methods canalso be applied to two-dimensional signals which need to be filtered bya two-dimensional sampling kernel and sampled by a two-dimensionalsampling system with a sampling rate at least equal to thetwo-dimensional rate of innovation.

The newly defined class of signals with finite rate of innovation pincludes a broad variety of signals of which the examples discussedabove are only a subset. The new sampling method can however be appliedto many signals x(t) with a finite rate of innovation p, regularlysampled so as to generate a set of sampled values y(nT) at a samplingfrequency f=1/T greater than said rate of innovation p, said sampledvalues y(nT) representing entirely the first signal x(t). Depending onthe form of the resulting equation system which only depends on theclass of signal to reconstruct, there can be an easy solution method forretrieving the shifts and weights.

In the above-described embodiments, the new sampling and reconstructionmethods are applied to sample and/or reconstruct a noiseless firstsignal x(t). The one skilled in the art will however recognize that inapplications with noisy conditions, the same methods can be used with ahigher sampling frequency, thus providing a higher number of sampledvalues x(nT). The corresponding equation systems described above canthen be solved using well-known spectral estimation techniques, such asfor example the singular value decomposition (SVD) method, in order toovercome the uncertainty induced by the noise.

In the above described embodiments, the sampled signal y_(s)[n] is usedfor reconstructing the signal y(t) before sampling, or at least a signalx(t) related to y(t) by a transfer function φ(t). The one skilled in theart will understand however that the sample values y_(s)[n], or even thesets of shifts t_(k) and weights c_(k) retrieved from those samplevalues, can be stored, transmitted and processed. In particular, a setof values y_(s1)[n] obtained by sampling a first signal y₁(t) at asub-Nyquist rate can be added to a second set of values y_(s2)[n]obtained by sampling a second signal y₂(t) at a sub-Nyquist rate. Therate of innovation ρ_(s) of the signal y_(s)(t)=y₁(t)+y₂(t) is obviouslythe sum of the rates of innovation ρ₁, ρ₂ of the summed signals y₁(t)and y₂(t). Therefore, for the sum of the sampled values to be asufficient representation of y_(s)(t), the number of samples of thesignal y_(s)(t) in a time interval must be greater than the sum of thenumber of degrees of freedom of the two added signals in this timeinterval. Therefore, if one wishes to perform some processing operationson the set of sampled values, one may increase the sampling rate inorder to have a sufficient number of samples in the processed signal.

The sampling method and the reconstruction method of the invention canbe performed by hardware circuits or systems and/or by a computerprogram product, including memories in which a software or a firmwarehas been stored and a computer program product directly loadable intothe internal memory of a digital processing system and comprisingsoftware code portions for performing the above described methods.Accordingly, the invention also relates to a circuit, to a system and toa computer program product adapted to carry out the above describedmethods, and to a processing circuit or program for processing sets ofvalues obtained by sampling a signal with the above described method.

1. A method implemented in an apparatus for reconstructing a firstsignal (x(t)), the method comprising: sampling a second signal (y(t)) ata sub-Nyquist rate and at uniform intervals; generating a set of sampledvalues (y_(s)[n], y(nT)) from the second signal (y(t)); retrieving fromsaid set of sampled values a set of shifts (t_(n), t_(k)) and weights(c_(n), c_(nr), c_(k)); and reconstructing the first signal (x(t)) basedon the set of shifts (t_(n), t_(k)) and weights (c_(n), c_(nr), c_(k)).2. Reconstruction method according to claim 1, wherein said set ofregularly spaced sampled values comprises at least 2K sampled values(y_(s)[n], y(nT)), wherein the class of said first signal (x(t)) isknown, wherein the bandwidth (B, |ω|) of said first signal (x(t)) ishigher than ω_(m)=π/T, T being the sampling interval, wherein the rateof innovation (ρ) of said first signal (x(t)) is finite, wherein saidfirst signal is faithfully reconstructed from said set of sampled valuesby solving a structured linear system depending on said known class ofsignal.
 3. Reconstruction method according to claim 1, wherein thereconstructed signal (x(t)) is a faithful representation of the sampledsignal (y(t)) or of a signal (x_(i)(t)) related to said sampled signal(y(t)) by a known transfer function (φ(t)).
 4. Reconstruction methodaccording to claim 3, wherein said transfer function (φ(t)) includes thetransfer function of a measuring device (7, 9) used for acquiring saidsecond signal (y(t)) and/or of a transfer channel (5) over which saidsecond signal (y(t)) has been transmitted.
 5. Reconstruction methodaccording to claim 1, wherein the reconstructed signal (x(t)) can berepresented as a sequence of known functions (γ(t)) weighted by saidweights (c_(k)) and shifted by said shifts (t_(k)).
 6. Reconstructionmethod according to claim 1, wherein the sampling rate is at least equalto the rate of innovation (ρ) of said first signal (x(t)). 7.Reconstruction method according to claim 1, wherein a first system ofequations is solved in order to retrieve said shifts (t_(k)) and asecond system of equations is solved in order to retrieve said weights(c_(k)).
 8. Reconstruction method according to claim 7, wherein theFourier coefficients (X[m]) of said sample values (y_(s)[n]) arecomputed in order to define the values in said first system ofequations.
 9. Reconstruction method according to claim 1, including thefollowing steps: fording at least 2K spectral values (X[m]) of saidfirst signal (x(t)), using an annihilating filter for retrieving saidarbitrary shifts (t_(n), t_(k)) from said spectral values (X[m]). 10.Reconstruction method according to claim 1, wherein said first signal(x(t)) is a periodic signal with a finite rate of innovation (ρ). 11.Reconstruction method according to claim 10, wherein said first signal(x(t)) is a periodical piecewise polynomial signal, said reconstructionmethod including the following steps: finding 2K spectral values (X[m])of said first signal (x(t)), using an annihilating filter for finding adifferentiated version (x^(R+1)(t)) of said first signal (x(t)) fromsaid spectral values, integrating said differentiated version to findsaid first signal.
 12. Reconstruction method according to claim 10,wherein said first signal (x(t)) is a finite stream of weighted Diracpulses$\left( {{x(t)} = {\sum\limits_{k = 0}^{K - 1}\; {c_{k}\; {\delta \left( {t - t_{k}} \right)}}}} \right),$said reconstruction method including the following steps: finding theroots of an interpolating filter to find the shifts (t_(n), t_(k)) ofsaid pulses, solving a linear system to find the weights (amplitudecoefficients) (c_(n), c_(k)) of said pulses.
 13. Reconstruction methodaccording to claim 1, wherein said first signal (x(t)) is a finitelength signal with a finite rate of innovation (ρ).
 14. Reconstructionmethod according to claim 13, wherein said reconstructed signal (x(t))is related to the sampled signal (y(t)) by a sin c transfer function(φ(t)).
 15. Reconstruction method according to claim 13, wherein saidreconstructed signal (x(t)) is related to the sampled signal (y(t)) by aGaussian transfer function (φ_(σ)(t)).
 16. Reconstruction methodaccording to claim 1, wherein said first signal (x(t)) is an infinitelength signal in which the rate of innovation (ρ, ρ_(T)) is locallyfinite, said reconstruction method comprising a plurality of successivesteps of reconstruction of successive intervals of said first signal(x(t)).
 17. Reconstruction method according to claim 16, wherein saidreconstructed signal (x(t)) is related to the sampled signal (y(t)) by aspline transfer function (φ(t)).
 18. Reconstruction method according toclaim 16, wherein said first signal (x(t)) is a bilevel signal. 19.Reconstruction method according to claim 16, wherein said first signal(x(t)) is a bilevel spline signal.
 20. Reconstruction method accordingto claim 1, wherein said first signal (x(t)) is a CDMA or a Ultra-WideBand signal.
 21. A method implemented in an apparatus for sampling afirst signal (x(t)), wherein said first signal (x(t)) can be representedover a finite time interval (τ) by the superposition of a finite number(K) of known functions (δ(t), γ(t), γ_(r)(t)) delayed by arbitraryshifts (t_(n), t_(k)) and weighted by arbitrary amplitude coefficients(c_(n), c_(k)), said method comprising: convoluting said first signal(x(t)) with a sampling kernel ((φ(t), φ(t)) and using a regular samplingfrequency (f, 1/T), choosing said sampling kernel ((φ(t), φ(t)) and saidsampling frequency (f, 1/T) such that sampled values (y_(s)[n], y(nT))completely specify said first signal (x(t)), and reconstructing saidfirst signal (x(t)), wherein said sampling frequency (f, 1/T) is lowerthan the frequency given by the Shannon theorem, but greater than orequal to twice said finite number (K) divided by said finite timeinterval (τ).
 22. Sampling method according to claim 21, wherein saidfirst signal (x(t)) is not bandlimited, and wherein said sampling kernel(φ(t)) is chosen so that the number of non-zero sampled values isgreater than 2K.